Complex Amplitude Modulation in Blood Flow and Perfusion Measurement

ABSTRACT

Systems and methods for blood flow and perfusion measurement using complex amplitude modulation of MRI pulses are presented. In exemplary embodiments of the disclosed subject matter, inflowing arterial spins can be modulated using a complex modulation function having certain mathematical properties in the frequency domain, such as, for example, a pseudo-random sequence. In exemplary embodiments of the disclosed subject matter the mathematical properties of such complex modulation functions can be used to measure individual transit times by deconvolving them from a series of acquired images. In exemplary embodiments of the disclosed subject matter images can be acquired at the same rapid rate as arterial modulation, and transit time distribution in the imaged tissue can be determined as part of a single integrated acquisition.

CROSS REFERENCE TO RELATED APPLICATIONS:

This application claims the benefit of U.S. Provisional PatentApplication No. 60/781,501, entitled “METHODS OF MEASURING FLOW ANDPERFUSION USING PSUEDO RANDOM AMPLITUDE MODULATION,” filed on Mar. 9,2006, the disclosure of which is hereby incorporated herein byreference.

TECHNICAL FIELD

The disclosed subject matter relates to medical imaging technology, andin particular to an improved method of measuring cerebral blood flow andperfusion using complex amplitude modulation schemes that allow for theresultant convolution of the distribution of transit times of thearterial blood with such modulation to be inverted by means of a Fourieranalysis of the observed signal at a position of interest.

BACKGROUND

Cerebral blood flow (“CBF”) is among the most important physiologicalvariables in understanding local brain metabolism. CBF provides thephysiological basis for blood oxygen level dependent (BOLD) contrast,the most frequently used imaging technique for estimating changes inneural activation¹ (all references cited to in this application arelisted in Appendix A hereto, which is incorporated herein by reference).In addition to its fluctuations during normal brain function, CBF alsochanges during many of the pathological events that lead to acute orchronic brain dysfunction. Because these pathological changes typicallyhave an earlier onset than structural biomarkers, measurement of CBF canbe a useful tool in diagnosing stroke, ischemia, brain tumors, anddementia.

The earliest measurements of CBF used radioactive tracers.² In fact,positron emission tomography (“PET”) measurements using radioisotopessuch as O¹⁵ are still regarded as the most accurate³. However, theinvasiveness and expense of radioactive tracers significantly inhibitthe widespread use of PET for CBF measurement.

Developments in MRI have led to an alternate method for measuring CBF,one using dynamic contrast enhancement following a rapid bolus injectionof MR contrast agent. Although less invasive and expensive thanradioactive tracers, this technique is still problematic, particularlyin research settings. The need for venipuncture discourages somepotential subjects, while others are excluded due to contraindicationsof the MRI contrast agent.

Arterial Spin Labeling (“ASL”) techniques replace the need for aninjected contrast agent by using water protons in the plasma as anendogenous MRI tracer. ASL is both non-invasive and cost effective, andhas higher spatial resolution than PET. Additionally, its clinicalutility is enhanced by the fact that ASL images can be routinelyacquired during the same imaging session as structural or other MRIscans and can be directly compared with the anatomical and pathologicalfeatures they reveal.

In ASL an endogenous tracer (arterial water) is used instead of anexogenous tracer. Flowing spins are inverted (labeled) at a plane in themain arteries which is proximal to an imaging volume. The labeled spinsthen flow through the arterial tree into the capillary bed arriving at aparticular location in the tissue with a distribution of arrival ortransit times. Once in the capillary bed, the arterial water moleculesthen exchange, one for one, with the extravascular water molecules. Dueto the accumulation of inverted spins in the extravascular component ofthe tissue there is a reduction in the total magnetization of thetissue. This, in turn, causes a reduction in the imaged signal in thatregion of the imaged plane. The reduction in signal of the tissue in theimaged plane is thus proportional to the amount of labeled spin whichflowed into that plane.

Using ASL, in general CBF can be measured by the subtraction of twoimages. The first is an unlabeled or control image in which there hasbeen no inversion of the spins. The subsequent image is labeled, i.e.,the spins have been inverted in an arterial plane proximal to the imagedslice. The resultant difference between the control and labeled imagesis directly proportional to the flow.

Two common methods of conducting ASL are Continuous ASL (“CASL”) andPulsed ASL (“PASL”). In PASL, spatially broad inversion pulses can beused to invert all of the spins in a region next to the slice ofinterest and the exchange between the inverted region and the imagedslice can be subsequently observed.⁹ In most cases PASL measures theratio of CBF at various locations because it is difficult to preciselydefine the amount of inflowing spins.

In CASL, incoming arterial water can be continuously inverted by anadiabatic inversion pulse. Although CASL can be used to measure CBF,this process is problematic in that it relies on certain assumptionsregarding the transit times and the relative T₁'s of blood and tissue todo so. If an image is acquired rapidly following an RF irradiation thelabeled arterial water will dominate the difference signal such that anysignal reduction from the water molecules that have exchanged into thetissue will not be visible. While this can be addressed by imposing apost labeling delay (PLD), for this to be successful the PLD needs to belonger than the longest transit time. This tends to make PLD's long,which results in lesser signal detection inasmuch as the differencesignal continually decreases as a function of PLD length.

In CASL, a separate acquisition is needed to obtain a control image.Flow can then be calculated from the difference between the controlimage and the labeled images. Transit times between the labeling andimaging planes are not measured in CASL. Nonetheless, transit time is animportant parameter in the calculation of quantitative CBF.¹⁴ Tominimize the role of this unknown variable in flow quantification,post-labeling delays (PLD) between the end of labeling and the start ofacquisition have been utilized, as noted. A PLD allows unlabeled spinswhich start flowing into the arterial tree sufficient time to wash outthe labeled spins so that any labeled spins left in the arteries willnot be confused with those that have exchanged into the tissue. Thus, ifthe PLD is longer than the transit time, the observed results areindependent of transit time provided that the T₁'s in the blood and thetissue are the same, which is approximately true for gray matter. Byvarying a PLD one can investigate the effects of various transit timessince an image acquired after a given PLD corresponds to integrationover all longer transit times.¹⁵

However, the insertion of a PLD has several disadvantages. First, it iscostly in SNR terms. Furthermore, it is vulnerable to changes in transitduring activation as well as the potentially longer transit times thatoften occur in stroke and other cerebral vascular diseases (which canexceed the PLD). For example, a standard PLD sufficiently small not todegrade SNR beyond acceptable limits is 1 to 2 seconds, and the transittimes common in stroke and other cerebral vascular diseases can be aslarge as 4 to 5 seconds.

In white matter, however, quantification of CBF using CASL encountersadditional unique difficulties. The use of a PLD for measuring CBF ingray matter depends upon the assumption that the T_(1gray)=T_(1blood),which, while acceptable for gray matter, is not valid for white matter.CASL also suffers from the lengthy transit time required for blood toreach white matter after being labeled in the artery. The relaxation ofthe labeled spins during this transit results in a much lower SNR forwhite matter relative to gray matter.

In ASL, the spins of protons in arterial water are typically inverted(labeled) at a plane in a cerebral artery. CBF then can be quantified bymeasuring the effect of the labeled spin when the protons enter a nearbyimaging plane. However, when the transit time between the labeling andimaging planes cannot be measured, a major uncertainty is introducedinto this calculation—it is difficult to discern how much of thedetected signal is due to label accumulated in the tissue or labelremaining in the arterial space in the tissue. While CASL attempts toreduce this uncertainty by introducing a PLD between the end of thelabeling pulse and the start of data acquisition, this solution is onlypartially effective; there is uncertainty regarding the relativerelaxation times, and it imposes an approximately three-fold cost insignal-to-noise ratio due to the longitudinal relaxation of the labelduring the PLD interval.

Moreover, current ASL methods use a continuous or square waveform forspin labeling. This can be difficult for the RF amplifiers on mostcommercial MR scanners to generate because they are generally designedfor pulsed, as opposed to continuous, applications.

Finally, CASL uses lengthy (e.g., 1-2 second) RF pulses for spinlabeling. The length of such pulses tends to place strain on the RFtransmitters in most MRI scanners that is typically relieved through theuse of transmit-receive head coils. However, receive-only coils offergreater sensitivity and are more widely available making this solutionnon-optimal. Additionally, long RF pulses introduce confounds frommagnetization transfer (MT) effects whereby the spins in the imagedplane are saturated by the long RF pulse. To address this, CASL requiresthat the acquisition of a control image be done in the presence of a RFpulse which mimics the MT effects of labeled images but does not invertthe arterial spins. However, while this approach eliminates MT effects,it introduces additional sources of error, particularly motionartifacts.

Thus, what is needed in the art is a method of blood flow and perfusionmeasurement that can overcome the problems of conventional methods, andthat can allow for the direct measurement of transit time distributionwhile acquiring ASL data.

What is further needed in the art is a method of blood flow andperfusion measurement that can add certainty to CBF quantification andobviate the need for PLD's control image acquisition or reliance on lesssensitive signal receiving hardware.

SUMMARY

Systems and methods for blood flow and perfusion measurement usingcomplex amplitude modulation of MRI pulses are presented. In exemplaryembodiments of the disclosed subject matter, inflowing arterial spinscan be modulated using a “complex” modulation function having definedmathematical properties in the frequency domain, such as, for example, apseudo-random sequence. In exemplary embodiments of the disclosedsubject matter such complex modulation functions can be used to measureindividual transit times by deconvolving them from a series of acquiredimages. In exemplary embodiments of the disclosed subject matter imagescan be acquired at the same rapid rate as arterial modulation, andtransit time distribution in the imaged tissue can be determined as partof a single integrated acquisition.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic representation of a pseudo-random amplitudemodulation scheme according to an exemplary embodiment of the disclosedsubject matter;

FIG. 2 is a schematic representation of pseudo-random sequencemodulation of inflowing arterial spins according to an exemplaryembodiment of the disclosed subject matter;

FIG. 3 is a schematic representation of arterial magnetization during anexemplary pseudo-random sequence according to an exemplary embodiment ofthe disclosed subject matter;

FIG. 4 illustrates the relationship between a PRAM inversion slab and animaging slice according to an exemplary embodiment of the disclosedsubject matter;

FIGS. 5(a)-(b) are images showing exemplary reconstructed transit timedistributions at 200 mL/min according to an exemplary embodiment of thedisclosed subject matter;

FIG. 6 depicts the results of a tagging experiment on a flow phantomaccording to an exemplary embodiment of the disclosed subject matter;

FIG. 7 depicts images showing reconstructed transit time distributionsat 100 mL/min according to an exemplary embodiment of the disclosedsubject matter;

FIGS. 8(a)-(b) depict PRAM images of a mouse brain according to anexemplary embodiment of the disclosed subject matter;

FIG. 9 is a schematic representation of a PRAM 180° pulse locationaccording to an exemplary embodiment of the disclosed subject matter;

FIG. 10 is a schematic representation of a PRAM 180° pulse locationinside a phase encode loop according to an exemplary embodiment of thedisclosed subject matter; and

FIG. 11 depicts an exemplary process flow according to an exemplaryembodiment of the disclosed subject matter.

DETAILED DESCRIPTION

Since their introduction in the 1990's,⁴ ⁵ ⁶ arterial spin labeling(“ASL”) techniques have proven to be very useful in acquiring absoluteand relative CBF measures, as well as regional perfusion data in a widevariety of circumstances.⁷ ASL techniques replace the need for aninjected contrast agent by using water protons in the plasma as anendogenous MRI tracer. ASL is both non-invasive and cost effective, andhas higher spatial resolution than PET. Additionally, its clinicalutility is enhanced by the fact that ASL images can be routinelyacquired during the same imaging session as structural or other MRIscans and can be directly compared with the anatomical and pathologicalfeatures they reveal.

In ASL an endogenous tracer (arterial water) is used instead of anexogenous tracer. Flowing spins are inverted (labeled) at a plane in themain arteries which is proximal to an imaging volume. The labeled spinsthen flow through the arterial tree into the capillary bed arriving at aparticular location in the tissue with a distribution of arrival ortransit times. Once in the capillary bed, the arterial water moleculesthen exchange, one for one, with the extravascular water molecules at arate given by KS, where K is the local permeation and S is the surfacearea of the capillary bed. Due to the accumulation of inverted spins inthe extravascular component of the tissue there is a reduction in thetotal magnetization of the tissue. This, in turn, causes a reduction inthe imaged signal in that region of the imaged plane. The reduction insignal of the tissue in the imaged plane is thus proportional to theamount of labeled spin which flowed into that plane.

In general, CBF can be measured by the subtraction of two images. Thefirst is an unlabeled or control image in which there has been noinversion of the spins. The subsequent image is labeled, i.e., the spinshave been inverted in an arterial plane proximal to the imaged slice.The resultant difference between the control and labeled images isdirectly proportional to the flow. If the inversion is modulated intime⁸ then the signal from the labeled spins flowing into the imagedslice is the convolution of the modulation function and the distributionof the transit times. This occurs because the observed signal at theimaged slice is the sum of static spins in the imaged plane togetherwith those brought to the imaged slice by the flow of the blood and itsperfusion into the tissue in the imaged slice. Each element of the sumis modulated by the amplitude of the modulation scheme employed to labelthe inflowing arterial blood. This sum is known as a convolution and canbe inverted to recover the distribution of the arterial transit times tothe imaged slice from the location of their inversion. This can bedescribed mathematically, for example, using the following equation:${S\left( {\overset{\rightarrow}{r},t} \right)} = {\int_{0}^{\infty}{{A(\tau)}{f\left( {\overset{\rightarrow}{r},{t - \tau}} \right)}\quad{\mathbb{d}\tau}}}$where S is the signal, A is the arterial modulation as a function oftime and f is the distribution of transit times.

As noted, two common methods of conducting ASL are continuous ASL (CASL)and pulsed ASL (PASL). In CASL, incoming arterial water is continuouslyinverted by an adiabatic inversion pulse. This pulse can be created bythe passage of the moving spins through a resonance condition created bya continuous RF waveform that irradiates the artery in the presence of agradient.⁴ Although CASL can be used to measure CBF, this process relieson certain assumptions regarding the transit times and the relative T₁'sof blood and tissue to do so. As described in greater detail below, ifan image is acquired rapidly following an RF irradiation the labeledarterial water will dominate the difference signal such that any signalreduction from the water molecules that have exchanged into the tissuewill not be visible. This can be addressed, for example, by imposing apost labeling delay (PLD) between the end of the RF irradiation and theacquisition of the image. For this to be successful, however, the PLDneeds to be longer than the longest transit time. This tends to makePLD's long, which results in lesser signal detection inasmuch as thedifference signal continually decreases as a function of PLD length.

In PASL spatially broad inversion pulses can be used to invert all ofthe spins in a region next to the slice of interest and the exchangebetween the inverted region and the imaged slice can be subsequentlyobserved.⁹ Details of how the inversions are done can, in general, varyconsiderably.^(5,10-12) In most cases PASL measures the ratio of CBF atvarious locations because it is difficult to precisely define the amountof inflowing spins. Recently, sequences have been developed that canaddress this problem¹³ by the addition of a separate RF pulse tosaturate inflowing spins after a certain time. Assuming a squarerectangular input to the imaged plane, these sequences can also be usedcan measure absolute CBF using similar assumptions as are made for CASL,as described above.

In standard CASL, a continuous RF waveform of resonant offset f₀ in thepresence of a gradient is used to invert all inflowing spins. Theinversion is achieved through an effective adiabatic pulse seen by thespins at the inversion plane z₀(f₀=yG z₀), since they pass from farbelow, through, and then above resonance as they move along the artery.The inverted spins then flow through an arterial tree and, after atransit time to reach the imaging plane, exchange with the tissue water.This method assumes that water is a freely diffusible tracer, which is areasonable assumption at most flow rates. The inversion pulse isgenerally left on for several seconds to ensure that the steady state ofmagnetization is reached prior to image acquisition.

In CASL, a separate acquisition is needed to obtain a control image.Flow can then be calculated from the difference between the controlimage and the labeled images. In general, the differences observedbetween labeled images and control images are on the order of 1%, which,although small, can be measured by averaging over several periods oflabel and control. One confound in this calculation is that therelatively long off-resonance continuous RF pulse used for labeling thespins can create magnetization transfer (MT) effects. These effectsresult from chemical exchange between protons in proteins and lipids andprotons in water, altering the characteristics of the labeled images bylowering the overall signal. Consequently, subtracting a control image(absence of RF) from a labeled image (presence of RF) includes MTeffects in addition to the desired flow information. Thus, thedifference image in such a case appears to be more different than itshould be due to the movement of the inverted spins into the imagedslice.

To circumvent this problem, a cosine amplitude modulation can beintroduced in the continuous waveform and applied during the acquisitionof the control image.^(14,15) Such a modulation creates two closelyspaced adiabatic inversion planes in the inflowing arterial spins sothat there is no net inversion. However, because the power levels of theRF waveforms for the control image and label image are adjusted to beequal, they induce similar MT effects in both. As a result these effectsare theoretically cancelled during the subtraction.

Transit times between the labeling and imaging planes are not measuredin CASL. Nonetheless, transit time is an important parameter in thecalculation of quantitative CBF.¹⁴ As noted above, to minimize the roleof this unknown variable in flow quantification, post-labeling delays(PLD) between the end of labeling and the start of acquisition can beutilized. A PLD allows unlabeled spins which start flowing into thearterial tree sufficient time to wash out the labeled spins so that anylabeled spins left in the arteries will not be confused with those thathave exchanged into the tissue. Thus, if the PLD is longer than thetransit time, the observed results are independent of transit timeprovided that the T₁'s in the blood and the tissue are the same, whichis approximately true for gray matter. By varying a PLD one caninvestigate the effects of various transit times since an image acquiredafter a given PLD corresponds to integration over all longer transittimes.¹⁵

Although the insertion of a PLD makes it possible for CASL to quantifyCBF without a priori knowledge of transit times, this solution hasseveral disadvantages. First, it is costly in SNR terms. Furthermore, itis vulnerable to changes in transit during activation as well as thepotentially longer transit times that often occur in stroke and othercerebral vascular diseases (which can exceed the PLD). For example, astandard PLD sufficiently small not to degrade SNR beyond acceptablelimits is 1-2 seconds, and the transit times common in stroke and othercerebral vascular diseases can be as large as 4 to 5 seconds.

In white matter, however, quantification of CBF using CASL encountersunique difficulties. The use of a PLD for measuring CBF in gray matterdepends upon the assumption that the T_(1gray)=T_(1blood), which, whileacceptable for gray matter, is not valid for white matter. CASL alsosuffers from the lengthy transit time required for blood to reach whitematter after being labeled in the artery. The relaxation of the labeledspins during this transit results in a much lower SNR for white matterrelative to gray matter.

As noted, CASL is a widely used technique for the measurement ofcerebral blood flow and brain perfusion in magnetic resonance imaging(“MRI”). Such measurement is clinically significant because alterationsin CBF occur early in many pathologies including stroke, tumor,ischemia, and dementia. Compared to other techniques for CBF imaging,CASL is non-invasive, inexpensive, and provides high-resolution images.It can also provide absolute quantification of CBF, but this potentialis rarely utilized due to the limitations of current methods asdescribed below.

In ASL, the spins of protons in arterial water are typically inverted(labeled) at a plane in a cerebral artery. CBF then can be quantified bymeasuring the effect of the labeled spin when the protons enter a nearbyimaging plane. As noted above, when the transit time between thelabeling and imaging planes cannot be measured, a major uncertainty isintroduced into this calculation—it is difficult to discern how much ofthe detected signal is due to label accumulated in the tissue or labelremaining in the arterial space in the tissue. CASL reduces thisuncertainty by introducing a post labeling delay (PLD) between the endof the labeling pulse and the start of data acquisition, provided thePLD is longer than the longest transit time and the relaxation times inthe tissue are similar to those in the artery. This means the signal isdominated by spins in the tissue rather than the arteries. However, thissolution is only partially effective; there is uncertainty regarding therelative relaxation times, and it imposes an approximately three-foldcost in signal-to-noise ratio due to the longitudinal relaxation of thelabel during the PLD interval.

Moreover, current ASL methods use a continuous or square waveform forspin labeling. This can be difficult for the RF amplifiers on mostcommercial MR scanners to generate because they are generally designedfor pulsed, as opposed to continuous, applications.

Finally, CASL uses lengthy (e.g., 1-2 second) RF pulses for spinlabeling. The length of such pulses tends to place strain on the RFtransmitters in most MRI scanners that is typically relieved through theuse of transmit-receive head coils. However, receive-only coils offergreater sensitivity and are more widely available making this solutionnon-optimal. Additionally, long RF pulses introduce confounds frommagnetization transfer (MT) effects whereby the spins in the imagedplane are saturated by the long RF pulse. To address this, CASL requiresthat the acquisition of a control image be done in the presence of a RFpulse which mimics the MT effects of labeled images but does not invertthe arterial spins. However, while this approach eliminates MT effects,it introduces additional sources of error, particularly motionartifacts.

CASL is currently the leading method for measuring CBF because it isnon-invasive, inexpensive, and compatible with anatomical and functionalMRI images acquired during the same scanning session. However, as noted,CASL is unable to measure transit times directly.

As noted above, while CASL can quantify CBF by introducing a PLD, such asolution is only partially effective for gray matter and less so forwhite matter. This is because the relaxation times in gray matter arecloser to those of arterial water than those of white matter. Further,because the transit times are longer for white matter than for graymatter, the larger difference in relaxation times is further amplified.Furthermore, a PLD reduces SNR and introduces confounds from MT.Additionally, because CASL cannot distinguish the effects of transittime from the measurement of flow, although it may detect a change, itcannot distinguish the source of that change. As it is widely recognizedthat some processes of interest, such as, for example, neural activationand stroke, may involve alterations of transit times as well as flow,this is a significant limitation.

Thus, in conventional ASL there is an absence of information regardingtransit time distribution. In exemplary embodiments of the disclosedsubject matter transit time distribution can be measured while acquiringASL data. This can be done, for example, by modulating the arterialinput with a “complex” modulation function, i.e., a function havingcertain properties in the frequency domain that allow the resultantconvolution of the distribution of transit times of the arterial bloodwith the complex modulation to be inverted by means of a Fourieranalysis of the observed signal at the position of interest. Thisrequires a modulation scheme whose Fourier transform has no zeros. Acommon example of such a complex modulation scheme is a pseudo randomsequence or “PRS.”

To illustrate such methods, an analysis of how the effects of thetransit time are incorporated into an acquired signal is next described.

As noted above, mathematically, a transit time distribution enters asits convolution with the arterial input modulation function. I.e., thesignal at each voxel is the integral of all of the spins from all of thedifferent transit times that traveled from the inversion plane to thatvoxel. This can be expressed, for example, as $\begin{matrix}{{\Delta\quad{I\left( {\overset{\rightarrow}{r},t} \right)}} = {\int_{0}^{\infty}{{A\left( {t - \tau} \right)}{f\left( {\overset{\rightarrow}{r},\tau} \right)}\quad{\mathbb{d}\tau}}}} & \left( {{Eq}.\quad A} \right)\end{matrix}$where ΔI is the change in image intensity observed at a specific pointand time due to the inflowing spins, A(t−τ) is the arterial spinamplitude modulation at the inversion plane at time t−τ, and f({rightarrow over (r)},τ) is the fraction of spins in a voxel that take τ timeunits to travel from the edge of the inversion slab to that voxel. It isnoted that this equation has been simplified by assuming that T₁ isinfinite in order to illustrate the effects of varying the arterialinput function. This assumption has the effect of ignoring anyrelaxation as the arterial spins travel from the inversion plane to theimaging plane. A more detailed description is provided below.

Barbier et al.^(8,16) have proposed a modulation scheme that is a squarewave of various frequencies which allow the time response of theperfusion to be mapped out. By acquiring images at different modulationfrequencies, the transit time dependence of the process can be measured.Thus, in the Barbier method A(t−τ) is a square wave of varyingfrequencies. In standard CASL the input function is a constant −1, butdue to waiting for a PLD delay time the effective signal is ∫_(PLD)^(∞)A(t−τ)f({right arrow over (r+EE,τ)dτ, where the lower limit of theintegral has been increased to the PLD inasmuch as subtraction of thecontrol image eliminates contributions from times shorter than the PLD.

To follow the motion of blood as it makes its way from a labeling siteto an imaging plane through an arterial tree, it is desirable to knowthe arterial transit time distribution function f({right arrow over(r)},τ), at each point in space and time. Theoretically, perhaps thesimplest way to do this is to invert a narrow band at the labeling siteand then observe those spins as they pass through an arterial tree andexchange into tissue across the capillary walls. In such case evaluatingthe convolution result is simple inasmuch as the arterial modulationfunction is effectively a delta function in time, A(t)=M₀δ(t₀), and thusI({right arrow over (r)},t)=M₀f({right arrow over (r)},t−t₀). Thisessentially states that a narrow inversion through an arterial systemcan be followed into the parenchyma, imaging it as it goes. In this waytransit times and accumulation in the parenchyma of the labeled spins ina series of sequential images can be measured. In practice, however,such a method does not work due to the fact that as the labeling getsnarrower intensity is lost and the duty cycle of the label becomes verysmall.

Thus, an alternative function for A is needed that can allow forsolution of the equation in real-world contexts. The nature ofalternative A's can be seen by recalling that if the Fourier transformof a convolution of two functions is taken, the product of the Fouriertransforms is obtained. Thus, if I=A conv. f, thenFT{I}=FT {A conv. f}=FT{A}*FT{f}, and FT{f}=FT{I}/FT{A}.

This fact makes the equation easy to solve for f, provided that thereare no zeros in the Fourier transform of A. In fact, this is why thedelta function works—it has unit amplitude everywhere in Fourier space.As is known, any function with unit amplitude in Fourier space must, byParseval's theorem, have a delta function as its autocorrelationfunction. What is therefore sought is a function that has no correlationwith itself over time, i.e., one that has a randomly modulated series of1's and −1's. One example of such a series is known as “pseudo-random”(also a maximum length) series because it resembles an uncorrelatedrandom series. However, since it has finite length, it cannot be trulyrandom and is thus termed “pseudo-random”. It is easy to generatepseudo-random sequences of length 2^(n)−1 because they can be, forexample, generated by linear shift registers based on prime binarypolynomials.¹⁷ If it is assumed to be such a sequence, then${{\sum\limits_{j = o}^{N - 1}\quad{AjAy}} - k} = {{{\left( {N + 1} \right)\delta_{ok}} - {1\quad{where}\quad N}} = {2^{n} - 1.}}$

In this expression the −1 comes from the fact that the number ofelements in the sequence is odd and thus cannot exactly cancel.

In exemplary embodiments of the disclosed subject matter these conceptscan be extended by modulating arterial spins with a complex functionthat satisfies these mathematical conditions. As noted above, apseudo-random sequence (PRS) is an example of such a complex function,and exemplary embodiments of the disclosed subject matter using a PRSwill be described in what follows. The extension to other complexmodulation functions is easily implemented as will be understood bythose skilled in the art. Such a modulation sequence can allow, forexample, for the transit time distribution to be deconvolved from theacquired image. Thus, expressed in the form of an equation, and assuminga PRS, the following can be obtained: $\begin{matrix}{{\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{k}} \right)}} = {{\int_{0}^{\infty}{{A\left( {t_{k} - t_{j}} \right)}{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}\quad{\mathbb{d}t_{j}}}} = {\sum\limits_{j = 0}^{2^{n} - 1}\quad{A_{k - j}{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}}}}} \\{{\sum\limits_{k = 0}^{2^{n} - 1}\quad{A_{k - l}\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{k}} \right)}}} = {\sum\limits_{k = 0}^{2^{n} - 1}\quad{A_{k - l}{\sum\limits_{j = 0}^{2^{n} - 1}\quad{A_{k - j}{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}}}}}} \\{= {\sum\limits_{j = 0}^{2^{n} - 1}\quad{\left( {\sum\limits_{k = 0}^{2^{n} - 1}\quad{A_{k - l}A_{k - j}}} \right){f\left( {\overset{\rightarrow}{r},t_{j}} \right)}}}} \\{= {{\sum\limits_{j = 0}^{2^{n} - 1}\quad{\left\{ {{\left( {2^{n} - 1} \right)\delta_{lj}} - 1} \right\}{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}}} \approx {\left( {2^{n} - 1} \right){f\left( {\overset{\rightarrow}{r},t_{l}} \right)}}}}\end{matrix}$(hereinafter the “Deconvolution Equation”).

In exemplary embodiments of the disclosed subject matter, arteriallabeling with a PRS can thus offer the advantages of CASL whileeliminating the problems associated with the introduction of a PLD.

In exemplary embodiments of the disclosed subject matter, an arterialinput function can be modulated with a PRS by applying a series of broadslice selective 180° RF pulses which inverts the spins in the entireregion between the chosen intput artery of interest and the region ofthe brain containing the imaged slice or slices of interest. The edge ofthe selective slice can be, for example, at a desired inversion planeand the width of the slice can be, for example, large enough to includethe imaging plane as well. The presence or absence of an inversion pulsecan be determined by the elements of the PRS, with a 180° pulse appliedwhenever the PRS changes sign, as shown, for example, in FIGS. 1 and 2(rows 140 in FIG. 1, and 250 in FIG. 2, respectively show the occurrenceof selective 180° pulses). This results in the longitudinalmagnetization of incoming arterial spins to be modulated to be equal tothe PRS.

In exemplary embodiments of the disclosed subject matter, the timebetween elements in such a series can be set to 100-200 ms so as to besufficiently short in comparison with variations in the transit timedistribution of approximately ˜1-2 seconds. Thus, in exemplaryembodiments of the disclosed subject matter, a PRS can, for example, becycled through at one element per TR. Thus, during each TR period newinflowing spins can cross the inversion plane and can be inverted or notaccording to the PRS modulation scheme.

In exemplary embodiments of the disclosed subject matter, an imagedslice can be sampled every TR. Assuming that a PRS has been exactlyreplicated, turning the original convolution integral (Eq. A above) intoa sum gives the following expression for the signal due to inflowingspins:${\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{n}} \right)}} = {{\sum\limits_{j = 0}^{N - 1}\quad{A_{n - j}{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}\quad{where}\quad t_{n}}} = {n \cdot {{TR}.}}}$

To invert this ΔI({right arrow over (r)},t_(n)) can be multiplied by ashifted A_(n), as follows: $\begin{matrix}{{\Delta\quad{S\left( {\overset{\rightarrow}{r},t_{k}} \right)}} = {{\sum\limits_{l = 0}^{N - 1}\quad{A_{k - l}\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{l}} \right)}}} = {\sum\limits_{l = 0}^{N - 1}\quad{A_{k - l}{\sum\limits_{j = 0}^{N - 1}{A_{l - j}{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}}}}}}} \\{= {{\sum\limits_{j = 0}^{N - 1}\quad{{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}{\sum\limits_{l = 0}^{N - 1}\quad{A_{k - l}A_{l - j}}}}} = {\sum\limits_{j = 0}^{N - 1}\quad{{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}{\sum\limits_{m = k}^{k - N + 1}\quad{A_{m}A_{k - j - m}}}}}}} \\{= {{\sum\limits_{j = 0}^{N - 1}{{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}\left( {{\left( {N + 1} \right)\delta_{{0\quad k} - j}} - 1} \right)}} = {{\left( {N + 1} \right){f\left( {\overset{\rightarrow}{r},t_{k}} \right)}} -}}} \\{{\sum\limits_{j = 0}^{N - 1}{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}} = {\left( {N + 1} \right)\left( {{f\left( {\overset{\rightarrow}{r},t_{k}} \right)} - \frac{\left\langle {f\left( \overset{\rightarrow}{r} \right)} \right\rangle}{N + 1}} \right)}} \\{\quad{{{where}{\quad}\left\langle {f\left( \overset{\rightarrow}{r} \right)} \right\rangle} = {\sum\limits_{j = 0}^{N - 1}{{f\left( {\overset{\rightarrow}{r},t_{j}} \right)}.}}}}\end{matrix}$

It is noted that $\begin{matrix}{{\sum\limits_{j = 0}^{N - 1}{\Delta\quad{S\left( {\overset{\rightarrow}{r},t_{j}} \right)}}} = {{\left( {N + 1} \right){\sum\limits_{j = 0}^{N - 1}\left( {{f\left( {\overset{\rightarrow}{r},t_{k}} \right)} - \frac{\left\langle {f\left( \overset{\rightarrow}{r} \right)} \right\rangle}{N + 1}} \right)}} =}} \\{{\left( {N + 1} \right)\left( {\left\langle {f\left( \overset{\rightarrow}{r} \right)} \right\rangle - \frac{\left\langle {f\left( \overset{\rightarrow}{r} \right)} \right\rangle}{N + 1}} \right)} = {N{\left\langle {f\left( \overset{\rightarrow}{r} \right)} \right\rangle.}}}\end{matrix}$

This allows for the recovery of the desired distribution function exceptfor a small correction due to the average transit time image <f({rightarrow over (r)})>, given by the two previous equations which can beeasily removed as a simple offset. In exemplary embodiments of thedisclosed subject matter, this offset can be calculated, for example, bysumming the ΔS's over the transit times. The images can then, forexample, be deconvolved with the input function to recover the images ofthe transit time distribution.

Thus, in exemplary embodiments of the disclosed subject matter, suchprocessing can effectively create images that are “snapshots” of wherethe spins have traveled to at that particular transit time.

In exemplary embodiments of the disclosed subject matter, by modulatingarterial spins with a such a complex sequence, such as, for example, aPRS, all of the advantages of CASL can be obtained without any of itsdrawbacks. Pseudo-random Amplitude Modulation (“PRAM”) can offer bothnon-invasive quantification of CBF as well as measurement of transittimes. This is possible because PRAM modulates the spins flowing acrossthe labeling plane according to a pseudo-random sequence. Subject to T₁recovery, the mathematical properties of this modulation of thelongitudinal magnetization in the arterial spins allow for measurementof all the transit times present in imaged tissue as part of a singleintegrated acquisition.

FIG. 1 schematically illustrates an exemplary modulation scheme A_(n)100 with a PRS of length 7 (L=2^(n)−1=3), according to an exemplaryembodiment of the disclosed subject matter. The desired magnetization110 is shown in the second row of the figure, a clock step TR 120 in thethird row and an image acquisition block 130 in the fourth row. Inexemplary embodiments of the disclosed subject matter, to achieve thedesired modulation of the incoming spins selective 180° pulses can, forexample, be applied as indicated in the bottom row 140 of FIG. 1. As canbe seen in FIG. 1, they occur each time the PRS changes sign. In thisexample, the entire sequence repeats after 7 steps, and in general a PRSrepeats after 2^(n)−1 steps.

The effect of such a series of 180° pulses is schematically illustratedin FIG. 2 for block flow in a tube. With reference thereto, top row 200illustrates the edge of a selective 180° pulse, while the bottom tworows show the desired magnetization profile 240, and the locations ofthe selective 180° pulses 250, respectively. Middle rows 210, 211, 220,221 and 230, 231 illustrate the effect of the RF pulses. In the exampleshown in FIG. 2, spins are flowing from left to right at velocity v, andare therefore moving a distance Δx=vΔt (Δt=TR) between clock pulses. Thespin distribution before pulse ‘d’ 210 shows a magnetization pattern of−1,1,−1,−1,1,1,1 (reading from right to left) which is what is expectedafter pulses ‘a’, ‘b’ and ‘c’0 have been applied. Thus, the effect ofpulse ‘d’ is simply to invert the existing distribution as shown in 211.In the next Δt more spins flow into the inversion slab, as shown in row220, and are then inverted after pulse ‘e’ as shown in row 221. Thisalso occurs between pulses ‘e’ and ‘f’, as shown in rows 230 and 231,respectively, leading to the final distribution which is the desiredmagnetization. It is noted that the rightmost spins in FIG. 2 willappear at the earliest times and thus are apparently reversed withrespect to the magnetization profile given in the figure.

A detailed theoretical analysis of the concepts underlying the disclosedsubject matter is next provided. To summarize, in exemplary embodimentsof the disclosed subject matter a sequence such as a PRS can be used tocreate a complex modulation of an input function at an inversion plane.This modulation can be determined by the chosen sequence and can beimplemented in the longitudinal magnetization of the arterial spins.These labeled spins can, for example, then travel to the imaged slice,lose their label at the arterial T_(a1), and exchange into theparenchyma through the capillaries. It is assumed here that all spinsexchange (using the known well mixed tissue assumption) and only leavethe tissue through the venous drainage at the rate set by the CBF. Toanalyze the effects of the PRS it can be assumed that TR<<T₁, and themodified Bloch equations can be linearized so that the magnetization canbe expressed in terms of the elements of the PRS, {A_(n)}.

As detailed below, there are two types of signals observed in the imagedslice—those that come from the static spins in the slice and those thatcome from the spins that exchange into the tissue from the capillaries.The signal from the static spins appears to have zero transit time anddecays with the combined effects of the tissue T₁, the CBF and theeffects of the excitation pulse. The signal from the exchanged spinsappears in the tissue after its transit time and subsequently decays atthe same rate as the static spins.

In what follows, the modulation of inflowing arterial spins caused bythe selective 180° pulses as controlled by a PRS is first analyzed.Next, the effects of the sequence of 180° pulses on the spins at theimaged slice, both static and flowing, are analyzed. In all cases it isassumed that a PRS is initially repeated once to prepare the system sothat the first pulse is not unique and the system is a steady state withrespect to the pseudo-random excitations. This means that sequence{A_(n)}, is cyclical and that the index should always be the calculatedmodulus of its length. It is understood that if a different complexmodulation scheme were employed then the detailed autocorrelationfunction of such employed modulation scheme would be used instead ofthat for a PRS.

To analyze the arterial input that appears at the tissue, it isnecessary to take into account all of the 180° pulses that the flowingspins experience. The magnetization that arrives at the imaged slice isthus a sum of the different transit times (due to different paths orvelocities) from the edge of the inversion slab to the slice location(subjected, of course, to any T1 relaxation that has occurred in theinterval). At every TR the arterial spins that have crossed into theinversion slab will or will not experience a 180° pulse as determined bythe PRS elements. Exactly how many of these they will experience dependson how long it takes them to travel to the slice location. For eachtransit time the arterial spins can be modulated differently by theeffects of the PRS. If the effects of relaxation are initially ignoredthen the equation relating the arterial longitudinal magnetization andthe flow distribution can be stated, for example, as $\begin{matrix}{{M_{i} = {\sum\limits_{j = 0}^{N - 1}\quad{{M^{0}\left( {- 1} \right)}^{\sum\limits_{k = i}^{i - j}\quad S_{k}}f_{j}}}},} & \lbrack 1\rbrack\end{matrix}$where M_(i) is the image observed at the i^(th) element in the PRS,f_(j) is the spins that have taken j TR periods to move from the edge ofthe inversion slab to the slice location, and S_(k) is either 0 or 1depending on whether there had been a 180° pulse at the k^(th) step ofthe PRS.

To extend this analysis to take into account the relaxation of thelabel, the Bloch equation,${\frac{\mathbb{d}{M_{a}(t)}}{\mathbb{d}t} = \frac{M_{a}^{0} - M_{a}}{T_{a\quad 1}}},$can be combined with the effects of the 180° pulses from the PRSsequence. Because of the variability of a PRS it is generally necessaryto consider the effect of each 180° pulse separately. This can be done,for example, by linearizing the Bloch equation (since TR<<T_(a1)), andcalculating the effects of each pulse recursively. FIG. 3 depicts anexemplary evolution of arterial magnetization M_(a) from t_(n−2) tot_(n+1) under the effects of the PRS {A_(n)} according to an exemplaryembodiment of the disclosed subject matter. Mathematically, a recursiverelationship can be developed for the arterial magnetization at timet_(n) in terms of a sum over variable transit times as follows, assumingthere is no stimulated echo formation which can be prevented by RF phasecycling in the PSR: $\begin{matrix}\begin{matrix}{{M_{a}\left( t_{n + 1} \right)} = {\left( {{M_{a}\left( t_{n} \right)} + {\Delta\quad{M_{a}\left( t_{n} \right)}}} \right)\left( {- 1} \right)^{S_{n}}}} \\{{\Delta\quad{M_{a}\left( t_{n} \right)}} = {\frac{M^{0} - {M_{a}\left( t_{n} \right)}}{T_{a\quad 1}}{TR}}} \\{{M_{a}\left( t_{n + 1} \right)} = {{\left\lbrack {{M^{0}\delta} + {{M_{a}\left( t_{n} \right)}\left( {1 - \delta_{a}} \right)}} \right\rbrack\left( {- 1} \right)^{S_{n}}\quad{where}\quad\delta_{a}} = \frac{TR}{T_{a\quad 1}}}} \\{= \left\lbrack {{M^{0}\delta_{a}} +} \right.} \\{\left. {\left\lbrack {{M^{0}\delta_{a}} + {{M_{a}\left( t_{n - 1} \right)}\left( {1 - \delta_{a}} \right)}} \right\rbrack\left( {- 1} \right)^{S_{n - 1}}\left( {1 - \delta_{a}} \right)} \right\rbrack\left( {- 1} \right)^{S_{n}}} \\{= \left\lbrack {{M^{0}\delta_{a}} + \left\lbrack {{M^{0}\delta_{a}} + \left\lbrack {{M^{0}\delta_{a}} +} \right.} \right.} \right.} \\{\left. {\left. {\left. {{M_{a}\left( t_{n - 2} \right)}\left( {1 - \delta_{a}} \right)} \right\rbrack\left( {- 1} \right)^{S_{n - 2}}\left( {1 - \delta_{a}} \right)} \right\rbrack\left( {- 1} \right)^{S_{n - 1}}\left( {1 - \delta_{a}} \right)} \right\rbrack\left( {- 1} \right)^{S_{n}}} \\{= {{M^{0}{{\delta_{a}\left( {- 1} \right)}^{S_{n}}\left\lbrack {1 + {\left( {- 1} \right)^{S_{n - 1}}\left( {1 - \delta_{a}} \right)\left\{ {1 + {\left( {- 1} \right)^{S_{n - 2}}\left( {1 - \delta_{a}} \right)}} \right\}}} \right\rbrack}} +}} \\{{M_{a}\left( t_{n - 2} \right)}\left( {1 - \delta_{a}} \right)^{3}\left( {- 1} \right)^{S_{n - 2} - S_{n - 1} - S_{n}}}\end{matrix} & \lbrack 2\rbrack\end{matrix}$so, in general, for a transit time of l

TR the following relationship between the magnetization at t_(n−1)results:${M_{a}\left( t_{n + 1} \right)} = {{{M_{a}\left( t_{n - l} \right)}\left( {1 - \delta_{a}} \right)^{l - 1}\left( {- 1} \right)^{\sum\limits_{j = 0}^{l}S_{n - j}}} + {M^{0}\delta_{a}{\sum\limits_{i = 0}^{l}{\left( {1 - \delta_{a}} \right)^{i}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j}}}}}}$

As noted above, S_(n) is cyclic. Thus n is the calculated modulus withrespect to the PRS period, N. The first term in equation [2] comes fromspins which have moved from the inversion plane to the imaging plane.The second term is a sum of all of the ΔM's accumulated in the lTRintervals that the blood experienced as it traveled to the imagingplane, each being modulated by the PRS pulses that occurred in theensuing time.${Further},{{\left( {1 - \delta_{a}} \right)^{i} \approx {\exp\left( {{- {\mathbb{i}}}\quad\delta_{a}} \right)}} = {{\exp\left( {- \frac{iTR}{T_{a\quad 1}}} \right)} = {{{\exp\left( {- \frac{t_{i}}{T_{a\quad 1}}} \right)}\quad{since}\quad{\exp\left( {- x} \right)}} = {\lim\limits_{l\rightarrow \propto}{\left( {1 - \frac{x}{i}} \right)^{i}\Phi}}}}}$

Thus, this equation reproduces the expected T_(a1) decay of the label asit travels through the arterial tree.

To convert this analysis to transit times, f(t_(j)) is set to representthe fraction of arterial magnetization that takes t_(j) to travel fromthe inversion plane to the imaging plane. The sum of f(t_(j)) over alltransit times is 1. The arterial magnetization at t_(n+1) is thus simplythe sum of the contributions from prior times multiplied by theirfractional contribution, which is given by f. $\begin{matrix}{{M_{a}\left( t_{n + 1} \right)} = {\sum\limits_{k = 0}^{N_{f}}{{f\left( t_{k} \right)}\left\{ {{{M_{a}\left( t_{n - k} \right)}\left( {1 - \delta_{a}} \right)^{k + 1}\left( {- 1} \right)^{\sum\limits_{j = 0}^{k}S_{n - j}}} + {M^{0}\delta_{a}{\sum\limits_{i = 0}^{k}{\left( {1 - \delta_{a}} \right)^{i}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j}}}}}} \right\}}}} & \lbrack 3\rbrack\end{matrix}$

where N_(f) is the maximum transit time (in TR's)$= {\sum\limits_{k = 0}^{N_{f}}{{f\left( t_{k} \right)}M^{0}\left\{ {{\left( {1 - \delta_{a}} \right)^{k + 1}\left( {- 1} \right)^{\sum\limits_{j = 0}^{k}S_{n - j}}} + {\delta_{a}{\sum\limits_{i = 0}^{k}{\left( {1 - \delta_{a}} \right)^{i}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j}}}}}} \right\}}}$or, readjusting the subscripts slightly${{M_{a}\left( t_{n} \right)} = {{M^{0}{\sum\limits_{k = 0}^{N_{f}}{{P_{n,k}\left( {1 - \delta_{a}} \right)}^{k - 1}{f\left( t_{k} \right)}}}} = {M^{0}{Pg}}}},{where}$${P_{n,k} = {\left( {- 1} \right)^{\sum\limits_{j = 0}^{k}S_{n - j - 1}} + {\delta_{a}{\sum\limits_{i = 0}^{k}{\left( {1 - \delta_{a}} \right)^{i - k - 1}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}}}}}},{g_{k} = {\left( {1 - \delta_{a}} \right)^{k + 1}{{f\left( t_{k} \right)}.}}}$

The step from the first to the second lines above follows asM_(a)(t_(n−k)) equals M⁰ because that group of spins crossed theinversion plane at t_(n−k) and is therefore then at their equilibriummagnetization. It is noted that the first part of the expression in [3]in the brackets is similar to equation [1]. In fact, in the limit asΔ→0, corresponding to an infinite T_(a1), the expressions are the same.For TR<<T_(a1) the second term in P_(n,k) can be ignored in comparisonwith the second term. Using this expression P can be calculated andinverted to get P⁻¹ which can, for example, then be used to solve theequation for f. Using this calculus transit time images for both exampleflow phantoms and a mouse brain were calculated, as described below.

The analysis can similarly be extended, for example, to perfused tissuein the imaged slice. In this case the situation is more complex becausethe effects of the excitation pulse have to be addressed and the factthat the arterial spins are exchanging into the tissue needs to be takeninto account. As is known, the Kety-Schmidt equation describes theexchange of water into tissue as given below in equation [4]. Thisassumes that water is a freely diffusible tracer and that the tissue canbe described as a single, well mixed solution.¹⁸ $\begin{matrix}{\frac{\mathbb{d}M_{t}}{\mathbb{d}t} = {{\frac{M_{t}^{0} - M_{t}}{T_{t\quad 1}} + {{CBF}\left\{ {M_{a} - \frac{M_{t}}{\lambda}} \right\}}} = {\frac{M_{t}^{0}}{T_{t\quad 1}} - {M_{t}\left( {\frac{1}{T_{t\quad 1}} - \frac{CBF}{\lambda}} \right)} + {{CBF} \cdot M_{a}}}}} & \lbrack 4\rbrack\end{matrix}$where M_(t) is the tissue magnetization, M_(t) ⁰ its equilibrium value,T_(t1) its longitudinal relaxation time, CBF the cerebral blood flow andλ the blood brain partition coefficient. It is noted that CBF is reallya function of position.

It is further noted that the relaxation time T_(t1) is much shorter (˜1sec) than the exchange lifetime CBF/Δ(˜100 sec); thus, the relaxationtime is the dominant pathway for the label to disappear.

To further analyze this equation, a similar linearizing approach can beused, for example, inasmuch as TR<<T₁. Also, the effects of theexcitation pulse on the spins, assumed to be φ, need to be accountedfor, in addition to the 180° pulses of the PRS. Otherwise the analysisis very similar. Thus, the detected signal can be expressed as M_(t) sinφ. $\begin{matrix}{{{M_{t}\left( t_{n + 1} \right)} = {\left( {{M_{t}\left( t_{n} \right)} + {\Delta\quad{M_{t}\left( t_{n} \right)}}} \right)\left( {- 1} \right)^{S_{n}}\cos\quad\phi}}{{\Delta\quad{M_{t}\left( t_{n} \right)}} = {{\frac{M_{t}^{0} - {M_{t}\left( t_{n} \right)}}{T_{t\quad 1}}{TR}} + {{CBF}\left\{ {{M_{a}\left( t_{n} \right)} - \frac{M_{t}\left( t_{n} \right)}{\lambda}} \right\}}}}{{M_{t}\left( t_{n + 1} \right)} = {\left\lbrack {{M_{t}^{0}\delta_{t}} + {{M_{t}\left( t_{n} \right)}\left( {1 - ɛ_{t}} \right)} + {{{CBF} \cdot {M_{a}\left( t_{n} \right)}}{TR}}} \right\rbrack\left( {- 1} \right)^{S_{n}}\cos\quad\phi}}{{{where}\quad\delta_{t}} = {{\frac{TR}{T_{t\quad 1}}\quad{and}\quad ɛ_{t}} = {\left( {\frac{1}{T_{t\quad 1}} + \frac{CBF}{\lambda}} \right){TR}}}}} & \lbrack 5\rbrack \\{{= {{\left\lbrack {{M_{t}^{0}\delta_{t}} + {\left\lbrack {{M_{t}^{0}\delta_{t}} + {{M_{t}\left( t_{n - 1} \right)}\left( {1 - ɛ_{t}} \right)} + {{{CBF} \cdot {M_{a}\left( t_{n - 1} \right)}}{TR}}} \right\rbrack\left( {- 1} \right)^{S_{n - 1}}\cos\quad{\phi\left( {1 - ɛ_{t}} \right)}} + {{{CBF} \cdot {M_{a}\left( t_{n} \right)}}{TR}}} \right\rbrack\left( {- 1} \right)^{S_{n}}\cos\quad\phi} = \begin{bmatrix}{{M_{t}^{0}\delta_{t}} + \begin{bmatrix}{{M_{t}^{0}\delta_{t}} + \left\lbrack {{M_{t}^{0}\delta_{t}} + {{M_{t}\left( t_{n - 2} \right)}\left( {1 - ɛ_{t}} \right)} + {{{CBF} \cdot {M_{a}\left( t_{n - 2} \right)}}{TR}}} \right\rbrack} \\{{\left( {- 1} \right)^{S_{n - 2}}\cos\quad{\phi\left( {1 - ɛ_{t}} \right)}} + {{{CBF} \cdot {M_{a}\left( t_{n - 1} \right)}}{TR}}}\end{bmatrix}} \\{{\left( {- 1} \right)^{S_{n - 1}}\cos\quad{\phi\left( {1 - ɛ_{t}} \right)}} + {{{CBF} \cdot {M_{a}\left( t_{n} \right)}}{TR}}}\end{bmatrix}}}{{\left( {- 1} \right)^{S_{n}}\cos\quad\phi} = {{M_{t}^{0}{\delta_{t}\left( {- 1} \right)}^{S_{n}}\cos\quad{\phi\left\lbrack {1 + {\left( {- 1} \right)^{S_{n - 1}}\left( {1 - ɛ_{t}} \right)\cos\quad\phi\left\{ {1 + {\left( {- 1} \right)^{S_{n - 2}}\left( {1 - ɛ_{t}} \right)\cos\quad\phi}} \right\}}} \right\rbrack}} + {{{CBF} \cdot {M_{t}\left( t_{n - 2} \right)}}\left( {1 - ɛ_{t}} \right)^{3}\cos^{3}{\phi\left( {- 1} \right)}^{S_{n - 2} + S_{n - 1} + S_{n}}} + {{CBF} \cdot {{TR}\left( {{{M_{a}\left( t_{n} \right)}\left( {- 1} \right)^{S_{n}}\cos\quad\phi} + {{M_{a}\left( t_{n - 1} \right)}\left( {- 1} \right)^{S_{n} + S_{n - 1}}\left( {1 - ɛ_{t}} \right)c\quad\text{?}\left( t_{n - 2} \right)\left( {- 1} \right)^{S_{n} + S_{n - 1} + S_{n - 2}}\left( {1 - ɛ_{t}} \right)^{2}\cos^{3}\phi}} \right)}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & \lbrack 5\rbrack\end{matrix}$after a total of N substitutions we have$= {{M_{t}^{0}\delta_{t}\cos\quad\phi{\sum\limits_{i = 0}^{N}{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j}}}}} + {{M_{t}\left( t_{n - N} \right)}\left( {1 - ɛ_{t}} \right)^{N + 1}\cos^{N + 1}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{N}S_{n - j}}} + {{{CBF} \cdot {TR} \cdot \cos}\quad\phi{\sum\limits_{i = 0}^{N}{{M_{a}\left( t_{n - i} \right)}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j}}\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}\phi}}}}$and the limit needs to be taken as N goes to infinity, or at least suchthat NTR>>T_(t1) or T_(a1), so as to reach a steady state. In such casethe middle term goes to zero because of (1−ε_(t))^(N+1). This then givesan equation for the magnetization in the tissue: $\begin{matrix}{{M_{t}\left( t_{n} \right)} = {{M_{t}^{0}\delta_{t}\cos\quad\phi{\sum\limits_{i = 0}^{N}{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}}}} +}} \\{{{CBF} \cdot {TR} \cdot}{\cos\quad\phi{\sum\limits_{i = 0}^{N}{{M_{a}\left( t_{n - i - 1} \right)}\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}}}}} \\{= {\cos\quad\phi{\sum\limits_{i = 0}^{N}\left\{ {{{CBF} \cdot {TR} \cdot {M_{a}\left( t_{n - i - 1} \right)}} + {M_{t}^{0}\delta_{t}}} \right\}}}} \\{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}}\end{matrix}$

Remembering that the detected signals is the magnetization before the φpulse times sin φ we find using equation [3] $\begin{matrix}{{{{M_{a}\left( t_{n} \right)} = {{M^{0}{\sum\limits_{k = 0}^{N_{f}}{{P_{n,k}\left( {1 - \delta_{a}} \right)}^{k - 1}{f\left( t_{k} \right)}}}} = {{M^{0}{Pg}\quad{where}\quad P_{n,k}} = {{\left( {- 1} \right)^{\sum\limits_{j = 0}^{k}S_{n - j - 1}} + {\delta_{a}{\sum\limits_{i = 0}^{k}{\left( {1 - \delta_{a}} \right)^{i - k - 1}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}\quad{and}\quad g_{k}}}}} = {\left( {1 - \delta_{a}} \right)^{k + 1}{f\left( t_{k} \right)}}}}}}\begin{matrix}{{S_{t}\left( t_{n} \right)} = {\sin\quad\phi\quad{\sum\limits_{i = 0}^{N}\left\{ {{{CBF} \cdot {TR} \cdot M_{a}^{0}}{\sum\limits_{k = 0}^{N_{f}}{P_{n - i - {1k}}\left( {1 - \delta_{a}} \right)}^{k + 1}}} \right.}}} \\{\left. {{f\left( t_{k} \right)} + {M_{t}^{0}\delta_{t}}} \right\}\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}} \\{= {\sin\quad\phi\quad M^{0}{\sum\limits_{i = 0}^{N}\left\{ {{{CBF} \cdot {TR} \cdot {\sum\limits_{k = 0}^{N_{f}}{{P_{n - i - {1k}}\left( {1 - \delta_{a}} \right)}^{k + 1}{f\left( t_{k} \right)}}}} + \delta_{t}} \right\}}}} \\{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\text{?}}}\end{matrix}{M_{a}\left( t_{n} \right)} = {{M^{0}{\sum\limits_{k = 0}^{N_{f}}{{P_{n,k}\left( {1 - \delta_{a}} \right)}^{k + 1}{f\left( t_{k} \right)}}}} = {{M^{0}{Pg}\quad{where}\quad P_{n,k}} = {{\left( {- 1} \right)^{\sum\limits_{j = 0}^{k}S_{n - j - 1}} + {\delta_{a}{\sum\limits_{i = 0}^{k}{\left( {1 - \delta_{a}} \right)^{i - k - 1}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}\quad{and}\quad g_{k}}}}} = {\left( {1 - \delta_{a}} \right)^{k + 1}{f\left( t_{k} \right)}}}}}}\begin{matrix}{{S_{t}\left( t_{n} \right)} = {\sin\quad\phi{\sum\limits_{i = 0}^{N}\left\{ {{{{CBF} \cdot {TR} \cdot M_{a}^{0}}{\sum\limits_{k = 0}^{N_{f}}{{P_{n - i - {1k}}\left( {1 - \delta_{a}} \right)}^{k + 1}{f\left( t_{k} \right)}}}} + {M_{t}^{0}\delta_{t}}} \right\}}}} \\{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}\quad{\varphi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}} \\{= {\sin\quad\phi\quad M^{0}{\sum\limits_{i = 0}^{N}\left\{ {{{CBF} \cdot {TR} \cdot {\sum\limits_{k = 0}^{N_{f}}{{P_{n - i - {1k}}\left( {1 - \delta_{a}} \right)}^{k + 1}{f\left( t_{k} \right)}}}} + \delta_{t}} \right\}}}} \\{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{{\phi\left( {- 1} \right)}^{\text{?}}.}}\end{matrix}{\text{?}\text{indicates text missing or illegible when filed}}} & \lbrack 6\rbrack\end{matrix}$

The solution has two parts. The first term in the brackets is due to thespins exchanging into the tissue from the arterial blood. The secondterm in the brackets is due to the static spins in the tissue. Becauseof the linearity of the system this can be analyzed by assuming thatf(t_(k))=δ_(mk), i.e., the only transit time is mTR. This gives$\begin{matrix}{{S_{t}\left( t_{n} \right)} = {\sin\quad\phi\quad M^{0}{\sum\limits_{i = 0}^{N}\left\{ {{{CBF} \cdot {TR}}{{{\cdot \left( {1 - \delta_{a}} \right)^{m + 1}}P_{n - i - {1m}}} + \delta_{t}}} \right\}}}} \\{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}} \\{= {\sin\quad\phi\quad M^{0}{\sum\limits_{i = 0}^{N}\left\{ {{CBF} \cdot {{TR}\left( {\left( {- 1} \right)^{\sum\limits_{j = 0}^{m}S_{n - i - j - 2}} + {\delta_{a}\sum\limits_{l = 0}^{m}}} \right.}} \right.}}} \\\left. {{\left. {\left( {1 - \delta_{a}} \right)^{l - m - 1}\left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - i - j - 2}}} \right)\left( {1 - \delta_{a}} \right)^{m + 1}} + \delta_{t}} \right\} \\{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}}\end{matrix}$

Ignoring the second term in the inner brackets, because it is smallrelative to the first, the following can be obtained: $\begin{matrix}\begin{matrix}\begin{matrix}{{S_{t}\left( t_{n} \right)} = {{in}\quad\phi\quad M^{0}{\sum\limits_{i = 0}^{N}\left\{ {{{{CBF} \cdot {TR} \cdot \left( {- 1} \right)^{\sum\limits_{j = 0}^{m}S_{n - i - j - 2}}}\left( {1 - \delta_{a}} \right)^{m + 1}} + \delta_{i}} \right\}}}} \\{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}} \\{= {\sin\quad\phi\quad M^{0}{\sum\limits_{i = 0}^{N}\left\{ {{{\delta_{t}\left( {1 - ɛ_{t}} \right)}^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}} +} \right.}}} \\{{{CBF} \cdot {TR} \cdot \left( {1 - \delta_{a}} \right)^{m + 1}}\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}{\phi\left( {- 1} \right)}^{\sum\limits_{j = 0}^{m}S_{n - i - j - 2}}} \\\left. \left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j - 1}} \right\} \\{\sin\quad\phi\quad M^{0}{\sum\limits_{i = 0}^{N}\left\{ {{\delta_{t}P_{n,i}G_{i}} + {{{CBF} \cdot {TR} \cdot \left( {1 - \delta_{a}} \right)^{m + 1}}P_{n,{i + m + 1}}G_{i}}} \right\}}}\end{matrix} \\{{{where}\quad G_{i}} = {{\left( {1 - ɛ_{t}} \right)^{i}\cos^{i}\quad\phi\quad{and}\quad P_{n,i}} = \left( {- 1} \right)^{\sum\limits_{j = 0}^{i}S_{n - j - 1}}}}\end{matrix} & \lbrack 7\rbrack\end{matrix}$

The static spins will appear in the transformed images at zero transittime and decay as (1−ε_(t))^(i) cos^(i) {tilde over (φ)}. The arterialspins that arrived at the tissue after a transit time of (m+1)TR, whichhave decayed only as (1−Δ_(a))^(m+1) and which were exchanged into thetissue will then also decay as (1−ε_(t))^(i) cos^(i) {tilde over (φ)}.Thus, two components are expected to be seen in the transit time images:(i) an initial component that decays to zero followed by (ii) a risingpeak that represents the CBF that subsequently decays. In exemplaryembodiments of the disclosed subject matter, by varying the excitationangle these decay rates can be controlled and the two parts of the curvecan be independently identified.

Determination of CBF

An important case occurs when φ=π/2. Then only the arrival of thearterial spins are seen since cos φ=0 and thus the only term in the sumover i is i=0. This provides an assumption free method of determiningCBF because the transit time distribution can be determined as nextdescribed.

Going back to equation [6], the following can be obtained:$\begin{matrix}{{S_{t}\left( t_{n} \right)} = {M^{0}\left\{ {{{CBF} \cdot {TR} \cdot {\sum\limits_{k = 0}^{N_{f}}{{P_{n - {1k}}\left( {1 - \delta_{a}} \right)}^{k + 1}{f\left( t_{k} \right)}}}} + \delta_{t}} \right\}\quad{Let}}} \\\begin{matrix}{{\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{l}} \right)}} = {\sum\limits_{j = 0}^{N - 1}{P_{ij}^{- 1}{S_{t}\left( t_{j + 1} \right)}}}} \\{= {\sum\limits_{j = 0}^{N - 1}{P_{lj}^{- 1}M^{0}\left\{ {{{CBF} \cdot {TR} \cdot {\sum\limits_{k = 0}^{N_{f}}{{P_{jk}\left( {1 - \delta_{a}} \right)}^{k + 1}{f\left( t_{k} \right)}}}} + \delta_{t}} \right\}}}} \\{= {M^{0}\left\{ {{{CBF} \cdot {TR} \cdot {\sum\limits_{j = 0}^{N - 1}{P_{lj}^{- 1}{\sum\limits_{k = 0}^{N_{f}}{{P_{jk}\left( {1 - \delta_{a}} \right)}^{k + 1}{f\left( t_{k} \right)}}}}}} +} \right.}} \\\left. {\sum\limits_{j = 0}^{N - 1}{P_{lj}^{- 1}\delta_{t}}} \right\}\end{matrix} \\\begin{matrix}{{\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{l}} \right)}} = {M^{0}\left\{ {{{{CBF} \cdot {TR} \cdot \left( {1 - \delta_{a}} \right)^{l + 1}}{f\left( t_{l} \right)}} + {\sum\limits_{j = 0}^{N - 1}{P_{lj}^{- 1}\delta_{t}}}} \right\}}} \\{= {{{M^{0} \cdot {CBF} \cdot {TR} \cdot \left( {1 - \delta_{a}} \right)^{l + 1}}{f\left( t_{l} \right)}} + {M^{0} \cdot \delta_{t} \cdot R_{l}}}}\end{matrix} \\{{{where}\quad R_{l}} = {\sum\limits_{j = 0}^{N - 1}P_{ij}^{- 1}}}\end{matrix}$

M⁰·δ can then be estimated form both long and short transit times whenf(t_(l)) is zero as R_(I) is known. Let it be Q, for example. Then anexpression for CBF can be calculated at each point r using the fact thatthe integral of f is 1.${M^{0} \cdot {{CBF}\left( \overset{\rightarrow}{r} \right)} \cdot {TR} \cdot {f\left( t_{l} \right)}} = {{{\frac{{\Delta\quad{I\left( {r,t_{l}} \right)}} - {QR}_{l}}{\left( {1 - \delta_{a}} \right)^{l + 1}}.\quad{Since}}\quad{\sum\limits_{l = 0}^{N_{f}}{f\left( t_{l} \right)}}} = 1}$${M^{0} \cdot {{CBF}\left( \overset{\rightarrow}{r} \right)} \cdot {TR}} = {\sum\limits_{l = 0}^{N_{f}}\frac{{\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{l}} \right)}} - {QR}_{l}}{\left( {1 - \delta_{a}} \right)^{l + 1}}}$Since we know M⁰=Q/δ_(t) we obtain the following formula for CBF:${{CBF}\left( \overset{\rightarrow}{r} \right)} = {{\frac{\delta_{t}}{Q \cdot {TR}}{\sum\limits_{l = 0}^{N_{f}}\frac{{\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{l}} \right)}} - {QR}_{l}}{\left( {1 - \delta_{a}} \right)^{l + 1}}}} = {\frac{1}{Q \cdot T_{t\quad 1}}{\sum\limits_{l = 0}^{N_{f}}\frac{{\Delta\quad{I\left( {\overset{\rightarrow}{r},t_{l}} \right)}} - {QR}_{l}}{\left( {1 - \delta_{a}} \right)^{l + 1}}}}}$in ml/sec/gr

This expression demonstrates the value of PRAM in ASL measurementsinasmuch as it shows the independence of the method to variations intransit time distribution. Of course, if the transit time is very slow,say 3 or 4 T_(a1)'s, then the division by (1−Δ_(a))^(I+1) becomesuntenable. Even for normal transit times this problem requires cuttingoff the sum at the point determined when the SNR is one, that is whenthe individual terms have reached the noise level in the particularexperiment in exemplary embodiments of the disclosed subject matter.

Using exemplary systems and methods according to the disclosed subjectmatter, the following experiments were performed.

In one experiment, images were acquired on an 89 mm vertical bore 9.4TBruker Biospin imager. A gradient echo sequence was used with a variableselective 180° pulse inserted immediately before the imaging block. The180° pulse, which inverted a 20 mm wide region, was either on or offdepending on the value of the element in a 63 element long PRS which waschosen to provide more than 3 seconds of length for a TR of 50 msec sothat slow velocities would be observable. The imaged slice was in themiddle of this region and was 1 mm thick. The relationship of theinversion slab and the imaged slice is depicted in FIG. 4 with d=1 cm,where flow 410 is depicted as running through inversion slab 420 fromleft to right in the figure. Imaging slice 430 was in the middle of theinversion slab, as noted. Imaging parameters were FOV 20 mm, TR 50, 100and 200 ms, matrix 64×64, and flip angle=45°. A total of 63×64transients were acquired, cycling through the PRS once for each k-spaceline. The flow phantom consisted of a 1.27cm inner diameter (i.d.) Tygontube containing water running through the RF coil of 3 cm i.d. Acomputer-controlled pump (Masterflex L/S) circulated the water in aclosed loop. Flows were varied from 50 to 200 mL/min.

At the flow rates used, flow was found to be non turbulent with aparabolic velocity distribution across the tube. The sequential transittime images of the flow distribution f after the inversion shown inequation [1] are displayed in FIG. 5 for TR=50 ms (FIG. 5(a)), andTR=100 ms (FIG. 5(b)), for a bulk flow of 200 mL/min, respectively. Theresults of a tagging experiment are shown in FIG. 6 (tags of 1.5 mmspacing, imaged after 200 ms) which demonstrates a parabolicdistribution. Inspection of FIG. 5 reveals that each later transit imageshows a wider and wider ring corresponding to a longer transit time.Comparison of the two different TRs shows similar patterns at the sametransit time delay. This can be seen, for example, by comparing thesecond image from the left in the bottom row of FIG. 5(a) (8^(th) imageor 400 ms) with the fourth image from the left in the top row of FIG.5(b) (4^(th) image or 400 ms). From the equations of Poisseuille flowthe theoretical maximum velocity in the center of the phantom can becalculated to be 5.26 cm/s, corresponding to a transit time of 190 ms.

In this experiment, several different flow rates and TR were alsoinvestigated. FIG. 7, for example, shows reconstructed transit timeimages of f for TR=100 ms for a flow of 100 mL/min, as compared withthose of FIG. 5 a obtained for 200 mL/min at TR=50 ms. In all cases theobserved transit time pattern was consistent with the theoreticalvelocities. As can be seen, the signal at the center of the tube in thetransit time images appears at the time predicted by the theory.Furthermore, the images clearly reflect the expected parabolic shape.

In addition to the flow phantom studies described above, experimentswere conducted on normal anaesthetized mice with a 1.6 cm inversion slabpositioned so that the inversion plane for the inflowing blood was 8 mmaway from the imaged slice. Data was acquired at TR=100 ms with both 45°and 90° excitation angles with a matrix of 128×128 and a FOV of 2.5 cmand 1 mm slice thickness. As in the flow phantom, the PRS was cycled ateach phase encode step. Acquisition time was approximately 15 minutes. Atypical result is shown in FIG. 8 which compares the reconstructedimages at the two angles for transit times of 0, 100, 200 and 300 ms,respectively. FIG. 8(a) depicts images acquired with Flip Angle=45°, andFIG. 8(b) depicts images acquired with Flip Angle=90°. As can be seenwith reference to FIG. 8, in both cases the first image has considerablestructure from the mouse brain and head. However, this structure almostcompletely disappears by 100 ms for the 90° pulse case while it is stillvisible at 300 ms for the 45° case. This result is completely consistentwith the theoretical predictions of equation [7] above, namely that thestatic spins will be visible only in the first reconstructed image for90° but will decay as (1−ε_(t))^(i) cos^(i) φ for f not equal to π/2.Also of note is the close agreement between the arterial brighteningseen in the two cases. As discussed in connection with equation [3]above, the arterial flow is expected to be relatively independent of thepulse angle because it only experiences the excitation pulse once(assuming that all flow is proximal to the imaged slice). These datashow that with the flexibility that PRAM provides it can be possible tocleanly separate static spins from flowing spins by varying theexcitation angle.

Thus, in exemplary embodiments of the disclosed subject matter, PseudoRandom Amplitude Modulation (PRAM) can be used to acquire transit timedata in ASL in a completely novel way. This enables PRAM to obtain CBFvalues without the problem of transit time effects. As described in thetheoretical analysis above, in exemplary embodiments of the disclosedsubject matter it is possible to directly estimate the local CBF with noassumptions beyond the arterial and tissue relaxation times. PRAM (aswell as other modulation functions or sequences satisfying the criteriaof “complexity” provided above) is able to do this because it candirectly measure the local transit time distribution arterialdistribution, essentially normalizing the arterial input function.Further, exemplary methods have a duty cycle of approximately 50% and donot require separate control and labeled images, as the entire sequenceis used to calculate the complete time course of the spins moving fromthe labeling plane to the imaging plane. Similar results can be achievedusing other complex modulation schemes.

Thus, in exemplary embodiments of the disclosed subject matter, anassumption free method of estimating the CBF at each point in the imagecan be implemented, removing the uncertainty regarding timedistributions that has hitherto made absolute quantitation difficult inASL.

In exemplary embodiments of the disclosed subject matter, a scanner canbe programmed to generate a desired complex modulation function, suchas, for example, a PRS, in conjunction with imaging sequences. Suchprograms can be, for example, written in a scanner optimized languagesuch as, for example, the Phillips programming code GOAL C forimplementation on a Phillips scanner, or using other known programminglanguages as may be desired.

In exemplary embodiments of the disclosed subject matter, an exemplaryprocess flow such as that depicted in FIG. 11 can be implemented with aseries of instructions to be executed by a computer or other dataprocessor. Such computer or other data processor can be, for example,integrated within an MRI machine and such series of instructions can beused to control such an MRI machine to implement methods according to anexemplary embodiment of the disclosed subject matter.

With reference to FIG. 11, such process flow will next be describedusing an exemplary PRS as a modulation function. At 1101, gradients andRF can be initialized from a field of view matrix and a labeling planelocation. From there process flow can, for example, move to 1105 wherean outer phase encode loop can be initialized (using an index “opx”) ifit is desired to image a 3D block of tissue rather than a single slice.Continuing at 1110, an inner phase encode loop can be initialized,using, for example, an index “ipx”. Process flow can then continue, forexample, to 1115 where a PRAM loop can be initialized, using, forexample, an index “PRS.”

At 1120, inversion gradient and RF on/off can be implemented accordingto the PRS index position in the PRS sequence. At 1125, image slice RFand gradient can be pulsed, at 1130, the phase encode gradients can bepulsed, and at 1135, for example, the gradient readout can be pulsed anda k-space line can be acquired. At 1140, for example, the PRS can beincremented and the system can then test for an end of loop. If an endof loop has been reached, process flow can move to 1145. If not, processflow can return, for example, to 1120. If process flow moves to 1145 at1140, then the ipx index (the inner phase encode loop index) can beincremented and the system can once again test for an end of loop. If ithas reached an end of loop, then process flow can, for example, move to1150. If no end of loop has been reached at 1145, then, for example,process flow can return to 1115. If process flow at 1145 has moved to1150, then an outer phase index (here “opx”) can be incremented, and thesystem can test for an end of loop. If at 1150 an end of loop has notbeen reached then process flow can, for example, return to 1110, and theprocess flow shown at 1110 through 1150 can repeat. If at 1150 an end ofloop in the outer phase encoding loop has been detected, then processflow can move to 1155 where raw data obtained via the processingdescribed above can be stored for analysis, and process flow can move to1157 where an inversion procedure can be implemented as, for example,described in the Deconvolution Equation presented above, to recoverdesired temporal distribution images. From 1157 process flow can, forexample, move to 1160 where it then ends.

Additionally, a more general loop structure can be implemented that canbe used with receive-only coils and standard gradient and imagingsequences in addition to EPI.

In an exemplary implementation according to the disclosed subjectmatter, a PRS can be relatively easily implemented by modifying astandard EPI sequence used to acquire TILT images¹², since such asequence already contains the needed 0°/180° RF and gradient pulses in aloop around the EPI acquisition code. Because the existing sequence onlyallows use of a transmit-receive (“T/R”) coil, this can be done, forexample, using a standard Phillips T/R head coil. The necessaryadditions to such a standard imaging sequence are highlightedschematically in FIG. 9. FIG. 9 depicts four rows, being RF 910, slice920, readout 930, and phase encode 940. Such additions can consist of,for example, inserting a selective RF pulse and gradient waveform at thebeginning of the sequence, which are contemplated and which are in acontrol loop that can modulate an RF pulse for either a 0° or 180° flipaccording to a given PRS sequence, as described above.

Schematically, an exemplary location of a new PRS controlled RF pulseand gradient waveform is illustrated in FIG. 9. As is illustrated inthis figure, the slice selection gradient does not have to be parallelin the PRAM inversion pulse and the imaged slice.

To implement the example implementation, code modifications can, forexample, involve altering the loop control structure for the existingTILT pulses so that it can handle 63 PRAM pulses. N=63 was chosenbecause of successful results found at 9.4T with this sequence. Inaddition, the correct dynamic scanning loops must be enabled since asingle pass through the PRS cycle will not have sufficient SNR. It isestimated that approximately 5 to 10 minutes of acquisition will beneeded to achieve adequate SNR. After acquisition, raw data can betransferred to, for example, a computer for analysis. Analysis canconsist of, for example, Fourier transforming the k-space data toproduce individual images measured at each point in the PRS cycle. Theseimages can then, for example, be deconvolved as described above.

In another exemplary implementation of the disclosed subject matter, acode module that can be inserted in a standard gradient echo imagingsequence can, for example, be developed. This can, for example,incorporate a PRS (or similar complex modulation sequence) loop insidethe phase encode loop as is illustrated in FIG. 10. FIG. 10 depicts thesame rows as are shown in FIG. 9, being RF 1010, slice 1020, readout1030, and phase encode 1040. Such a code module can include, forexample, insertion of the PRS loop and generation of appropriatevalidation code required for the scanner to operate within the safetylimits set by the FDA. This can be done, for example, by adaptation of acardiac phase cycle loop which already exists inside the phase encodeloop. This can, for example, enable the PRS to repeat once for eachvalue of the phase encode gradient needed to generate the desired image.As before, the dynamic loop structures must be enabled to allow multipleacquisitions. Once such a module has been developed for gradient echosequences it can, for example, be extended to spin echo sequences byinserting appropriate 180° pulses during the readout. The extra pulsescan be accounted for in the theory by balancing the readout gradientsand the use of 180°-φ pulses to excite the imaged slice so that the neteffect is still only that of a φ pulse since a 360° pulse does notexcite the spins in any way. Raw data can be saved directly as it isacquired and later transferred to, for example, a computer for analysis.

In exemplary embodiments of the disclosed subject matter a prospectivePRAM sequence can be thoroughly tested and validated using a flowphantom consisting of both stationary and flowing spins. To do this, forexample, a cylindrical sample (i.d. 10 cm) with a continuous tube (i.d.2 cm) through its center can be constructed. The tube can be connectedto a variable speed pump so that the flow can be adjusted. The maximumspeed in the center of the tube can range, for example, from 1 to 20cm/sec.

As noted above, a PRAM procedure depends on the TR, the PRS length (n),the distance from the edge of the inversion slab to the imaged plane,the excitation pulse angle, the slice thickness, and the flow rate. Therelationship among the TR, distance between the inversion plane and theimaged slice, and the flow speed is simply v=d/t.

Thus, PRAM images can be acquired at five different flows, correspondingto maximum velocities of 1, 2, 5, 10 and 20 cm/s. The FOV can be, forexample, 128 by 128 mm acquired with a 64×64 matrix, leading to 2 mmin-plane resolution. This can provide sufficient resolution over thetube to resolve the expected parabolic velocity profile, i.e. highestintensity at the center of the image. An initial PRS, {A_(n)}, used canbe 63 elements long, generated by a linear shift register using, forexample, the polynomial, x⁶+x+1, and can be, for example:{A_(n)}=−1,1,−1,1,−1,1,−1,−1,1,1,−1,−1,1,−1,−1,−1,1,−1,−1,1,−1,1,1,−1,1,1,−1,−1,−1,1,1,1,−1,1,−1,−1,−1,−1,1,1,−1,1,−1,1,1,1,−1,−1,1,1,1,1,−1,1,1,1,1,1,−1,−1,−1,−1,−1}.

Each image acquired can, for example, be stored as raw k-space data andsubsequently transferred to a computer or data processor, such as forexample, a PowerEdge™ 6800, for further processing using, for example,Matlab programs written to carry out the matrix inversions describedabove. If a longer PRS is needed then a prime polynomial of seventhdegree (x⁷+x+1) can be used, for example, to generate a 127 elementsequence. A longer sequence may, for example, be needed to study veryshort TR's since the product of PRS length and TR needs to be longerthan the difference in transit time between the fastest and slowestvelocities in order to prevent overlap between the slowest velocitiesfrom the previous PRS cycle and the fastest velocities from the presentcycle.

At each flow rate (for example, 1, 2, 5, 10, 20 cm/s) the TR can bevaried, such as, for example, as 50, 100, 150, 200 and 250 ms.Additionally, for example, the distance from the edge of the inversionslab to the imaged slice (2, 5 and 10 cm), and the slice excitationangle (20°, 45° and 90°) and imaged slice thickness (2 and 10 mm) can bevaried. To keep the number of separate acquisitions reasonable, theparameters can typically be varied over the indicated range at, forexample, one or two sets of values of the remaining parameters.

In addition to PRAM acquisitions at each flow rate, standard spintagging can also be used to directly measure the actual velocities inthe tube. Furthermore, CASL images with five post labeling delays andthe inversion plane set at the edge of the PRAM inversion slab can beacquired. The shortest PLD can, for example, be chosen to allow thelabeled spins to clear at the maximum velocity (so that the center oftube will just be unlabeled). Four additional acquisitions can, forexample, each have their PLD increased by a factor of 3^(1/4) in orderto acquire of a range of images with the labeled spins cleared fromincreasing larger radii. With this choice of multiplicative factor thelongest PLD should correspond to a velocity found at a diameter of 1.8cm assuming a parabolic velocity distribution. These images can, forexample, then be compared to the sum of the transformed PRAM resultsfrom the PLD onward in transit times. Images from the CASL sequences andthe summed transformed PRAM images can be compared by examining profilesto determine the radii where the labeled spins vanish at each postlabeling delay.

Using exemplary systems and methods according to the disclosed subjectmatter, the following described experiments could, for example, be run.

A first exemplary experiment could be run to confirm that a PRAMsequence is producing expected data. Early transit time images producedby PRAM sequences can be tested for congruence with an arterial treeobserved by the very well validated TOF MRA procedure. For example, sixvolunteers can participate in such an experiment. During two scanningsessions, a standard 3D T1 sequence at an isotropic resolution of 1.5 mmfor anatomical reference can be acquired. Then 20 5 mm thick slices witha PRAM sequence verified as above be acquired. Since these acquisitionsare designed to detect the signal from the arterial blood which is muchhigher than the signal from perfusion, they will need only a 1 minuteacquisition for adequate S/N. Allowing 30 seconds for repositioning theslice, we expect that acquisition of the entire dataset will require atotal of 30 minutes. Each slice can be moved superiorly so that a 10 cmthick block will ultimately be covered. The block can be positioned sothe uppermost slice will be at the top of the brain. The raw data can betransformed and deconvolved to generated transit time images of eachslice. The same region of the brain can then be imaged with standard TOFMRA and reconstructed using maximum intensity projection (MIP) algorithmon the scanner. The reconstructed MIP images can also be transferred toa computer. Both datasets can be examined using a 3D visualizationprogram, for example, Almira, to confirm that the expected blood flowmotion through the arterial tree observed by PRAM visually coregistersthe detailed arterial anatomy seen on the MEP projections. Sixvolunteers can be used in order to sample a range of variations innormal anatomy.

In a second exemplary experiment, the results of CBF measurement couldbe, for example, obtained with PRAM and with CASL. Twelve volunteers canparticipate in such an experiment. Separate scans performed during twoscanning sessions will cover different slice positions and willalternate between PRAM and CASL. PRAM data can be, for example, acquiredwith a 90° pulse to minimize the contributions from the static tissueand implement the CBF calculation procedure of equation [8]. Thesevalues can be compared to the CBF values obtained by CASL using thestandard formulae of Alsop¹⁴. Further, the same slice can be acquired atseveral values of the excitation angle, as equation [7] shows that thetrailing shape of the transit time curve in the tissue parenchyma iscontrolled by the decay rate ε but the uptake depends on the transittime distribution. In this way the static spin contributions can beclearly separated from the inflowing ones which contribute to thetransit time distribution.

Because it is expected that the spread in transit times will be lessthan 3 seconds¹⁴, the same 63 PRS can be used initially. Initial studiescan, for example, be conducted with TR equal to 100 ms, although it willbe possible to use a TR as small as 50 ms for a PRS cycle time of 3.15seconds. High resolution, 3DT1 images (SPGR) can be acquired first foranatomical reference. The PRS cycle can be repeated 100 times in thePRAM sequence to increase SNR, resulting in a total acquisition time of10.5 minutes. Three excitation pulse angles can be used, for example,20°, 45°, and 90°. FOV and matrix can be 240×240 mm and 64×64 for aresolution of 3.75 mm. CASL sequences can be acquired with the same FOVand matrix and 5 different post labeling delays (400, 600, 800, 1000 and1200 ms). Each acquisition can be 5 minutes, for a total of 20 minutes.

A voxel-wise comparison of the calculated flow for CASL for each PLD andPRAM according to the formula in equation [8] can, for example, then beobtained. It should be noted that for CASL, the CBF will vary with PLDreflecting the overall integral of the transit time larger than the PLDchosen.

The methods and systems of exemplary embodiments of the disclosedsubject matter thus provide an inexpensive and non-invasive MRItechnique for the diagnosis of diseases including dementia, stroke,ischemia, and brain tumors through the detection of pathological changesin cerebral blood flow that occur early in the pathogenesis of theseconditions.

Moreover, such methods and systems can further contribute to thedevelopment of magnetic resonance imaging techniques that can be usedfor quantitative measurement of cerebral blood flow, which will benefitresearchers and clinicians studying the brain.

While this invention has been described with reference to one or moreexemplary embodiments thereof, it is not to be limited thereto and theappended claims are intended to be construed to encompass not only thespecific forms and variants of the invention shown, but to furtherencompass such as may be devised by those skilled in the art withoutdeparting from the true scope of the invention. It is further understoodthat the various disclosed embodiments can be combined, rearranged,substituted, adapted, etc. as may be convenient or desirable, all suchcombinations, rearrangements, substitutions, adaptations, etc., beingwithin the spirit and scope of the invention.

Appendix A REFERENCES

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1. A method for measuring fluid flow and transit times in a sample viamagnetic resonance imaging, comprising: providing a labeling plane andan imaging plane for a sample; modulating the magnetization of inflowingspins at the labeling plane using a complex sequence; and obtainingtransit times for the modulated spins from the labeling plane to theimaging plane from a set of acquired image slices of the sample in asingle unified acquisition, wherein the complex sequence has minimalautocorrelation over time and its frequency domain transform has nozeros, and wherein the transit times are obtained by dividing thefrequency domain transform of the acquired signal by the frequencydomain transform of the complex sequence.
 2. The method of claim 1,wherein the complex sequence is a pseudo random sequence having 2^(n)−1elements, where n is an integer.
 3. The method of claim 2, wherein thepseudo random sequence comprises one of 63 and 127 elements.
 4. Themethod of claim 2, wherein the pseudo random sequence is cycled throughonce for each k-space line.
 5. The method of claim 2, wherein the pseudorandom sequence is used to modulate a 180° variable selective pulseapplied to the sample.
 6. The method of claim 1, wherein the complexsequence is used to modulate a 180° variable selective pulse applied tothe sample.
 7. The method of claim 1, wherein the sample is an artery,and wherein arterial longitudinal magnetization and flow distributionare related according to:${M_{i} = {\sum\limits_{j = 0}^{N - 1}{{M^{0}\left( {- 1} \right)}^{\sum\limits_{k = i}^{i - j}S_{k}}f_{j}}}},$wherein M_(i) is the image observed at the i^(th) element in the pseudorandom sequence, f_(j) is the spins that have taken j TR periods to movefrom the edge of an inversion slab to an imaging slice location, andS_(k) is 0 or 1 depending on whether there had been a 180° pulse at ak^(th) step of the pseudo random sequence.
 8. The method of claim 7,wherein the complex sequence is a pseudo random sequence comprising2^(n−)1 elements, where n is an integer.
 9. The method of claim 2,wherein the number of image slices of the sample acquired equals thelength of the pseudo-random sequence.
 10. A system for measuring transittimes and cerebral blood flow in an artery via magnetic resonanceimaging, comprising: an MRI scanner; a data processor; and a controller,wherein in operation the controller executes a series of instructionswhich cause the MRI scanner to: provide a labeling plane and an imagingplane for a sample; modulate the magnetization of inflowing spins at thelabeling plane using a complex sequence; and causes the data processorto: process the acquired signal at the imaging plane to obtain transittimes for the modulated spins from the labeling plane to the imagingplane from a set of acquired image slices of the sample in a singleunified acquisition, wherein the complex sequence has minimalautocorrelation over time and its frequency domain transform has nozeros, and wherein the transit times are obtained by dividing thefrequency domain transform of the acquired signal by the frequencydomain transform of the complex sequence.
 11. The system of claim 10,wherein the complex sequence is a pseudo-random sequence having 2^(n)−1elements, where n is an integer.
 12. The system of claim 11, wherein thepseudo-random sequence comprises one of 63 and 127 elements.
 13. Thesystem of claim 11, wherein the pseudo-random sequence is cycled throughonce for each k-space line.
 14. The system of claim 11, wherein thepseudo random sequence is used to modulate a 180° variable selectivepulse applied to the sample.
 15. The system of claim 10, wherein thecomplex sequence is used to modulate a 180° variable selective pulseapplied to the sample.
 16. The system of claim 15, wherein the complexsequence is a pseudo random sequence comprising 2^(n)−1 elements, wheren is an integer.
 17. The system of claim 11, wherein the sample is anartery, and wherein arterial longitudinal magnetization and flowdistribution are related according to:${M_{i} = {\sum\limits_{j = 0}^{N - 1}{{M^{0}\left( {- 1} \right)}^{\sum\limits_{k = i}^{i - j}S_{k}}f_{j}}}},$wherein M_(i) is the image observed at the i^(th) element in the pseudorandom sequence, f_(j) is the spins that have taken j TR periods to movefrom the edge of an inversion slab to an imaging slice location, andS_(k) is 0 or 1 depending on whether there had been a 180° pulse at ak^(th) step of the pseudo random sequence.
 18. The system of claim 11,wherein the number of image slices of the sample acquired equals thelength of the pseudo-random sequence.
 19. The system of claim 10,wherein the number of image slices of the sample acquired equals thelength of the complex sequence.
 20. A computer program productcomprising a computer usable medium having computer readable programcode means embodied therein, the computer readable program code means insaid computer program product comprising means for causing a computerto: provide a labeling plane and an imaging plane for a sample; modulatethe magnetization of inflowing spins at the labeling plane using acomplex sequence; and obtain transit times for the modulated spins fromthe labeling plane to the imaging plane from a set of acquired imageslices of the sample in a single unified acquisition; wherein thecomplex sequence has minimal autocorrelation over time and its frequencydomain transform has no zeros, and wherein the transit times areobtained by dividing the frequency domain transform of the acquiredsignal by the frequency domain transform of the complex sequence. 21.The computer program product of claim 20, wherein the complex sequenceis a pseudo-random sequence having 2^(n)−1 elements, where n is aninteger.
 22. The computer program product of claim 21, wherein thepseudo random sequence comprises one of 63 and 127 elements.
 23. Thecomputer program product of claim 21, wherein the pseudo random sequenceis cycled through once for each k-space line.
 24. The computer programproduct of claim 21, wherein the pseudo random sequence is used tomodulate a 180° variable selective pulse applied to the sample.
 25. Thecomputer program product of claim 21, wherein the sample is an artery,and wherein arterial longitudinal magnetization and flow distributionare related according to:${M_{i} = {\sum\limits_{j = 0}^{N - 1}{\left( {M^{0}\left( {- 1} \right)} \right)^{\sum\limits_{k = i}^{i - j}S_{k}}f_{j}}}},$wherein M_(i) is the image observed at the i^(th) element in the pseudorandom sequence, f_(j) is the spins that have taken j TR periods to movefrom the edge of an inversion slab to an imaging slice location, andS_(k) is 0 or 1 depending on whether there had been a 180° pulse at ak^(th) step of the pseudo random sequence.
 26. The computer programproduct of claim 21, wherein the number of image slices of the sampleacquired equals the length of the pseudo-random sequence.
 27. Thecomputer program product of claim 20, wherein the complex sequence isused to modulate a 180° variable selective pulse applied to the sample.